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Complex quadratic polynomial

From Wikipedia, the free encyclopedia

A complex quadratic polynomial is a quadratic polynomial whose coefficients and variable are complex numbers.

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  • ✪ Example: Complex roots for a quadratic | Algebra II | Khan Academy
  • ✪ Precalculus 12.1a - Complex Quadratic
  • ✪ Factoring a Complex Quadratic
  • ✪ Quadratic inequalities | Polynomial and rational functions | Algebra II | Khan Academy
  • ✪ Quadratic equations with complex coefficients


We're asked to solve 2x squared plus 5 is equal to 6x. And so we have a quadratic equation here. But just to put it into a form that we're more familiar with, let's try to put it into standard form. And standard form, of course, is the form ax squared plus bx plus c is equal to 0. And to do that, we essentially have to take the 6x and get rid of it from the right hand side. So we just have a 0 on the right hand side. And to do that, let's just subtract 6x from both sides of this equation. And so our left hand side becomes 2x squared minus 6x plus 5 is equal to-- and then on our right hand side, these two characters cancel out, and we just are left with 0. And there's many ways to solve this. We could try to factor it. And if I was trying to factor it, I would divide both sides by 2. If I divide both sides by 2, I would get integer coefficients on the x squared in the x term, but I would get 5/2 for the constant. So it's not one of these easy things to factor. We could complete the square, or we could apply the quadratic formula, which is really just a formula derived from completing the square. So let's do that in this scenario. And the quadratic formula tells us that if we have something in standard form like this, that the roots of it are going to be negative b plus or minus-- so that gives us two roots right over there-- plus or minus square root of b squared minus 4ac over 2a. So let's apply that to this situation. Negative b-- this right here is b. So negative b is negative negative 6. So that's going to be positive 6, plus or minus the square root of b squared. Negative 6 squared is 36, minus 4 times a-- which is 2-- times 2 times c, which is 5. Times 5. All of that over 2 times a. a is 2. So 2 times 2 is 4. So this is going to be equal to 6 plus or minus the square root of 36-- so let me just figure this out. 36 minus-- so this is 4 times 2 times 5. This is 40 over here. So 36 minus 40. And you already might be wondering what's going to happen here. All of that over 4. Or this is equal to 6 plus or minus the square root of negative 4. 36 minus 40 is negative 4 over 4. And you might say, hey, wait Sal. Negative 4, if I take a square root, I'm going to get an imaginary number. And you would be right. The only two roots of this quadratic equation right here are going to turn out to be complex, because when we evaluate this, we're going to get an imaginary number. So we're essentially going to get two complex numbers when we take the positive and negative version of this root. So let's do that. So the square root of negative 4, that is the same thing as 2i. And we know that's the same thing as 2i, or if you want to think of it this way. Square root of negative 4 is the same thing as the square root of negative 1 times the square root of 4, which is the same. I could even do it one step-- that's the same thing as negative 1 times 4 under the radical, which is the same thing as the square root of negative 1 times the square root of 4. And the principal square root of negative 1 is i times the principal square root of 4 is 2. So this is 2i, or i times 2. So this right over here is going to be 2i. So we are left with x is equal to 6 plus or minus 2i over 4. And if we were to simplify it, we could divide the numerator and the denominator by 2. And so that would be the same thing as 3 plus or minus i over 2. Or if you want to write them as two distinct complex numbers, you could write this as 3 plus i over 2, or 3/2 plus 1/2i. That's if I take the positive version of the i there. Or we could view this as 3/2 minus 1/2i. This and these two guys right here are equivalent. Those are the two roots. Now what I want to do is a verify that these work. Verify these two roots. So this one I can rewrite as 3 plus i over 2. These are equivalent. All I did-- you can see that this is just dividing both of these by 2. Or if you were to essentially factor out the 1/2, you could go either way on this expression. And this one over here is going to be 3 minus i over 2. Or you could go directly from this. This is 3 plus or minus i over 2. So 3 plus i over 2. Or 3 minus i over 2. This and this or this and this, or this. These are all equal representations of both of the roots. But let's see if they work. So I'm first going to try this character right over here. It's going to get a little bit hairy, because we're going to have to square it and all the rest. But let's see if we can do it. So what we want to do is we want to take 2 times this quantity squared. So 2 times 3 plus i over 2 squared plus 5. And we want to verify that that's the same thing as 6 times this quantity, as 6 times 3 plus i over 2. So what is 3 plus i squared? So this is 2 times-- let me just square this. So 3 plus i, that's going to be 3 squared, which is 9, plus 2 times the product of three and i. So 3 times i is 3i, times 2 is 6i. So plus 6i. And if that doesn't make sense to you, I encourage you to kind of multiply it out either with the distributive property or FOIL it out, and you'll get the middle term. You'll get 3i twice. When you add them, you get 6i. I And then plus i squared, and i squared is negative 1. Minus 1. All of that over 4, plus 5, is equal to-- well, if you divide the numerator and the denominator by 2, you get a 3 here and you get a 1 here. And 3 distributed on 3 plus i is equal to 9 plus 3i. And what we have over here, we can simplify it just to save some screen real estate. 9 minus 1 is 8. So if I get rid of this, this is just 8 plus 6i. We can divide the numerator and the denominator right here by 2. So the numerator would become 4 plus 3i, if we divided it by 2, and the denominator here is just going to be 2. This 2 and this 2 are going to cancel out. So on the left hand side, we're left with 4 plus 3i plus 5. And this needs to be equal to 9 plus 3i. Well, you can see we have a 3i on both sides of this equation. And we have a 4 plus 5, which is exactly equal to 9. So this solution, 3 plus i, definitely works. Now let's try 3 minus i. So once again, just looking at the original equation, 2x squared plus 5 is equal to 6x. Let me write it down over here. Let me rewrite the original equation. We have 2x squared plus 5 is equal to 6x. And now we're going to try this root, verify that it works. So we have 2 times 3 minus i over 2 squared plus 5 needs to be equal to 6 times this business. 6 times 3 minus i over 2. Once again, a little hairy. But as long as we do everything, we put our head down and focus on it, we should be able to get the right result. So 3 minus i squared. 3 minus i times 3 minus i, which is-- and you could get practice taking squares of two termed expressions, or complex numbers in this case actually-- it's going to be 9, that's 3 squared, and then 3 times negative i is negative 3i. And then you're going to have two of those. So negative 6i. So negative i squared is also negative 1. That's negative 1 times negative 1 times i times i. So that's also negative 1. Negative i squared is also equal to negative 1. Negative i is also another square root. Not the principal square root, but one of the square roots of negative 1. So now we're going to have a plus 1, because-- oh, sorry, we're going to have a minus 1. Because this is negative i squared, which is negative 1. And all of that over 4. All of that over-- that's 2 squared is 4. Times 2 over here, plus 5, needs to be equal to-- well, before I even multiply it out, we could divide the numerator and the denominator by 2. So 6 divided by 2 is 3. 2 divided by 2 is 1. So 3 times 3 is 9. 3 times negative i is negative 3i. And if we simplify it a little bit more, 9 minus 1 is going to be-- I'll do this in blue. 9 minus 1 is going to be 8. We have 8 minus 6i. And then if we divide 8 minus 6i by 2 and 4 by 2, in the numerator, we're going to get 4 minus 3i. And in the denominator over here, we're going to get a 2. We divided the numerator and the denominator by 2. Then we have a 2 out here. And we have a 2 in the denominator. Those two characters will cancel out. And so this expression right over here cancels or simplifies to 4 minus 3i. Then we have a plus 5 needs to be equal to 9 minus 3i. I We have a negative 3i on the left, a negative 3i on the right. We have a 4 plus 5. We could evaluate it. This left hand side is 9 minus 3i, which is the exact same complex number as we have on the right hand side, 9 minus 3i. So it also checks out. It is also a root. So we verified that both of these complex roots, satisfy this quadratic equation.



Quadratic polynomials have the following properties, regardless of the form:


When the quadratic polynomial has only one variable (univariate), one can distinguish its four main forms:

  • The general form: where
  • The factored form used for logistic map
  • which has an indifferent fixed point with multiplier at the origin[2]
  • The monic and centered form,

The monic and centered form has been studied extensively, and has the following properties:

The lambda form is:

  • the simplest non-trival perturbation of unperturbated system
  • "the first family of dynamical systems in which explicit necessary and sufficient conditions are known for when a small divisor problem is stable"[4]


Between forms

Since is affine conjugate to the general form of the quadratic polynomial it is often used to study complex dynamics and to create images of Mandelbrot, Julia and Fatou sets.

When one wants change from to :[5]

When one wants change from to the parameter transformation is[6]

and the transformation between the variables in and is

With doubling map

There is semi-conjugacy between the dyadic transformation (the doubling map) and the quadratic polynomial case of c = –2.



Here denotes the n-th iteration of the function (and not exponentiation of the function):


Because of the possible confusion with exponentiation, some authors write for the nth iterate of the function


The monic and centered form can be marked by:

  • the parameter
  • the external angle of the ray that lands:
    • at c in M on the parameter plane
    • at z = c in J(f) on the dynamic plane

so :


The monic and centered form, sometimes called the Douady-Hubbard family of quadratic polynomials,[7] is typically used with variable and parameter :

When it is used as an evolution function of the discrete nonlinear dynamical system

it is named the quadratic map:[8]

The Mandelbrot set is the set of values of the parameter c for which the initial condition z0 = 0 does not cause the iterates to diverge to infinity.

Critical items

Critical point

A critical point of is a point in the dynamical plane such that the derivative vanishes:



we see that the only (finite) critical point of is the point .

is an initial point for Mandelbrot set iteration.[9]

Critical value

A critical value of is the image of a critical point:


we have

So the parameter is the critical value of

Critical orbit

Dynamical plane with critical orbit falling into 3-period cycle
Dynamical plane with critical orbit falling into 3-period cycle
Dynamical plane with Julia set and critical orbit.
Dynamical plane with Julia set and critical orbit.
Dynamical plane : changes of critical orbit along internal ray of main cardioid for angle 1/6
Dynamical plane : changes of critical orbit along internal ray of main cardioid for angle 1/6
Critical orbit tending to weakly attracting fixed point with abs(multiplier)=0.99993612384259
Critical orbit tending to weakly attracting fixed point with abs(multiplier)=0.99993612384259

The forward orbit of a critical point is called a critical orbit. Critical orbits are very important because every attracting periodic orbit attracts a critical point, so studying the critical orbits helps us understand the dynamics in the Fatou set.[10][11][12]

This orbit falls into an attracting periodic cycle if one exists.

Critical sector

The critical sector is a sector of the dynamical plane containing the critical point.

Critical polynomial


These polynomials are used for:

  • finding centers of these Mandelbrot set components of period n. Centers are roots of n-th critical polynomials

Critical curves

Critical curves
Critical curves

Diagrams of critical polynomials are called critical curves.[13]

These curves create the skeleton (the dark lines) of a bifurcation diagram.[14][15]

Spaces, planes

4D space

One can use the Julia-Mandelbrot 4-dimensional ( 4D) space for a global analysis of this dynamical system.[16]

w-plane and c-plane
w-plane and c-plane

In this space there are 2 basic types of 2-D planes:

  • the dynamical (dynamic) plane, -plane or c-plane
  • the parameter plane or z-plane

There is also another plane used to analyze such dynamical systems w-plane:

  • the conjugation plane[17]
  • model plane[18]

2D Parameter plane

Gamma parameter plane for complex logistic map  z n + 1 = γ z n ( 1 − z n ) , {\displaystyle z_{n+1}=\gamma z_{n}\left(1-z_{n}\right),}
Gamma parameter plane for complex logistic map
Multiplier map
Multiplier map

The phase space of a quadratic map is called its parameter plane. Here:

is constant and is variable.

There is no dynamics here. It is only a set of parameter values. There are no orbits on the parameter plane.

The parameter plane consists of:

There are many different subtypes of the parameter plane.[20][21]

See also :

  • Boettcher map which maps exterior of mandelbrot set to the exterior of unit disc
  • multiplier map which maps interior of hyperbolic component of Mandelbrot set to the interior of unit disc

2D Dynamical plane

 "The polynomial Pc maps each dynamical ray to another ray doubling the angle (which we measure in full turns, i.e. 0 = 1 = 2π rad = 360◦), and the dynamical rays of any polynomial “look like straight rays” near infinity. This allows us to study the Mandelbrot and Julia sets combinatorially, replacing the dynamical plane by the unit circle, rays by angles, and the quadratic polynomial by the doubling modulo one map." Virpi K a u k o[22]

On the dynamical plane one can find:

The dynamical plane consists of:

Here, is a constant and is a variable.

The two-dimensional dynamical plane can be treated as a Poincaré cross-section of three-dimensional space of continuous dynamical system.[23][24]

Dynamical z-planes can be divided in two groups :

  • plane for ( see complex squaring map )
  • planes ( all other planes for )

Riemann sphere

The extended complex plane plus a point at infinity


First derivative with respect to c

On the parameter plane:

  • is a variable
  • is constant

The first derivative of with respect to c is

This derivative can be found by iteration starting with

and then replacing at every consecutive step

This can easily be verified by using the chain rule for the derivative.

This derivative is used in the distance estimation method for drawing a Mandelbrot set.

First derivative with respect to z

On the dynamical plane:

  • is a variable;
  • is a constant.

At a fixed point

At a periodic point z0 of period p the first derivative of a function

is often represented by and referred to as the multiplier or the Lyapunov characteristic number. Its logarithm is known as the Lyapunov exponent. It used to check the stability of periodic (also fixed) points.

At a nonperiodic point, the derivative, denoted by can be found by iteration starting with

and then using

This derivative is used for computing the external distance to the Julia set.

Schwarzian derivative

The Schwarzian derivative (SD for short) of f is:[25]


See also


  1. ^ Alfredo Poirier : On Post Critically Finite Polynomials Part One: Critical Portraits
  2. ^ Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials. 
  3. ^ Bodil Branner: Holomorphic dynamical systems in the complex plane. Mat-Report No 1996-42. Technical University of Denmark
  4. ^ Dynamical Systems and Small Divisors, Editors: Stefano Marmi, Jean-Christophe Yoccoz, page 46
  5. ^ Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials. 
  6. ^ stackexchange questions : Show that the familiar logistic map ...
  7. ^ Yunping Jing : Local connectivity of the Mandelbrot set at certain infinitely renormalizable points  Complex Dynamics and Related Topics, New Studies in Advanced Mathematics, 2004, The International Press, 236-264
  8. ^ Weisstein, Eric W. "Quadratic Map." From MathWorld--A Wolfram Web Resource
  9. ^ Java program by Dieter Röß showing result of changing initial point of Mandelbrot iterations Archived 26 April 2012 at the Wayback Machine
  10. ^ M. Romera Archived 22 June 2008 at the Wayback Machine, G. Pastor Archived 1 May 2008 at the Wayback Machine, and F. Montoya : Multifurcations in nonhyperbolic fixed points of the Mandelbrot map. Archived 11 December 2009 at the Wayback Machine Fractalia Archived 19 September 2008 at the Wayback Machine 6, No. 21, 10-12 (1997)
  11. ^ Burns A M : Plotting the Escape: An Animation of Parabolic Bifurcations in the Mandelbrot Set. Mathematics Magazine, Vol. 75, No. 2 (Apr., 2002), pp. 104-116
  12. ^ Khan Academy : Mandelbrot Spirals 2
  13. ^ The Road to Chaos is Filled with Polynomial Curves by Richard D. Neidinger and R. John Annen III. American Mathematical Monthly, Vol. 103, No. 8, October 1996, pp. 640-653
  14. ^ Hao, Bailin (1989). Elementary Symbolic Dynamics and Chaos in Dissipative Systems. World Scientific. ISBN 9971-5-0682-3. Archived from the original on 5 December 2009. Retrieved 2 December 2009.
  15. ^ M. Romera, G. Pastor and F. Montoya, "Misiurewicz points in one-dimensional quadratic maps", Physica A, 232 (1996), 517-535. Preprint Archived 2 October 2006 at the Wayback Machine
  16. ^ Julia-Mandelbrot Space at Mu-ENCY (the Encyclopedia of the Mandelbrot Set) by Robert Munafo
  17. ^ Carleson, Lennart, Gamelin, Theodore W.: Complex Dynamics Series: Universitext, Subseries: Universitext: Tracts in Mathematics, 1st ed. 1993. Corr. 2nd printing, 1996, IX, 192 p. 28 illus., ISBN 978-0-387-97942-7
  18. ^ Holomorphic motions and puzzels by P Roesch
  19. ^ Lasse Rempe, Dierk Schleicher : Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity
  20. ^ Alternate Parameter Planes by David E. Joyce 
  21. ^ exponentialmap by Robert Munafo
  22. ^ Trees of visible components in the Mandelbrot set by Virpi K a u k o , FUNDAM E N TA MATHEMATICAE 164 (2000)
  23. ^ Mandelbrot set by Saratov group of theoretical nonlinear dynamics
  24. ^ Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia,
  25. ^ The Schwarzian Derivative & the Critical Orbit by Wes McKinney  18.091  20 April 2005

External links

This page was last edited on 27 September 2019, at 18:23
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